What makes you say that?
What is the middle game plan here?

To my eye, the problem with playing d4 in that position is that you undermine your control of e4. After a trade on d4, Black gets to break with d5 and your f-pawn gets undermined.
I think the engine's point about Be3 and using the bishop to recapture on d4 is to control the e5 square so that after Black's d5 break, you don't have to go exd5 but can instead push e4-e5 and f5-f6. That's a nice idea and one I wouldn't have caught.
What other ideas come to mind? My first thought was to leave the center closed and try to take advantage of the f5-pawn by a maneuver like Nh4 plus Ne2-g3-h5 (since they can't really play g7-g6). But that may be too slow -- they do have d5 ideas, and even if they're less effective with White's pawn still on d3, they may be effective enough to stop a maneuver that long. And you're right that it may just come to nothing if they don't castle or castle long.
Maybe there are g4-g5-g6 ideas, but that's extremely risky of course.
Nerves of Butter, I don't disagree but the criticism seems ill-placed. The OP seems to understand that human ideas are needed and is asking for them. He's *not* asking to be spoon-fed correct moves. Rather, he's explaining what limitations he's had in thinking about the position strategically and is wondering what he's missing. You're telling him to do exactly what he's trying to do -- except you don't seem to want him to ask anyone for their thoughts, which is exactly what the forums are for.

What on earth are you talking about? He gave like ten times more thoughts on what's going on in the position that most people do when they post. This is not someone asking to be spoon-fed. Please do not discourage good questions. Save it for when people really do ask bad ones.
@1
"In this position I played d4"
++ In this position white is winning.
Black has weakened his position and lost a tempo with 6...h6?
The white pawn structure is better. Bc1 is a good bishop, not on the color of its pawns.
Bc4 is a bad bishop, but it is outside of its pawn chain c2-d3-e4-f5 and thus not bad at all.
Be7 is a bad bishop, on the color of its own pawns d6 and e5.
Bc8 is his good bishop, not on the color of its pawns, but it stares at pawn f5.
So there is no advantage in playing d4: it only worsens white's position.
On the other hand black would benefit from playing ...d5, freeing his bishops.
So white should prevent ...d5 and develop pieces.
Hence 11 Qe2 is good: keeping pawn c2 protected and creating a vis a vis of Qe2 and Ke8 and further putting pressure on central pawn e5 and clearing the path to bring Ra1 to d1.
After developing Qe2, Be3, Rad1 and maybe the prophylactic Bb3 to overprotect pawn c2 and to not lose a tempo in case of ...d5 you can think of attacking g4 to exploit the weakness created by 6...h6?
"it was a blitz game" ++ 'He who analyses blitz is stupid.' - Nezhmetdinov
"What if he doesn't castle?" ++ Then his king will be in danger in the center.
"How will I attack him then?" ++ First complete your development: Qe2, Be3, Rad1, Bb3
"d4 immediately gives up all the advantage"
++ Do not play the same piece twice in the opening. You have played the pawn to d3.
That is fine. Now complete development: Qe2, Be3, Rad1, Bb3.
"the engine wants my piece on d4 to be protected twice (by the queen and by the knight on f3)" ++ Overprotection is good, as Nimzovich said.
"what is the correct middle plan for white here"
++ Complete your development, then attack with g4

I'm not sure how to explain this succinctly. Opening the center when an opponent hasn't castled is a good idea. However, you don't have any pressure in the center because you're not fully developed yourself. Furthermore, you don't want the center to be opened if black castles to the kingside because you have so much space there already, opening the center just gives black more squares and lines to maneuver their pieces to the defense. Thus, the best idea is to finish development then if black castles keep the center closed and attack on the kingside and if black keeps his king in the center you will be poised to play d4 at some moment when it is now too late for black to castle. I think this position is very hard for black and the best policy is patience.
In this position I played d4 trying to open up the center which computer says is a mistake. I understand that when the position is closed my space advantage on the kingside and the fact that his pieces are cramped benefit me but it was a blitz game and I started thinking "What if he doesn't castle? How will I attack him then? I must open the center!" so I panicked and played d4. The interesting thing is that had I played Be3 and my opponent responded with almost any move other than castles then the engine says d4 would be completely fine, while d4 immediately gives up all the advantage. I explored why that is and I saw that after d4 when white plays exd4 the engine wants to take back with the bishop and in the event of opening the d file the engine wants my piece on d4 to be protected twice (by the queen and by the knight on f3). All of this made me think what is the correct middle plan for white here, in the position that you see? The best move is Be3, at least according to my engine, but I want to know what is the correct plan for white in this position.
Here is the fen of the position, for some reason I can't post it as a picture of a chess board.
r1bqk2r/1p2bpp1/2pp1n1p/p3pP2/PnB1P3/2NP1N2/1PP3PP/R1BQ1R1K w kq - 0 11
Edit: Just skip to the end to see the position that I am talking about. I am not sure if I replicated the game correctly, I just made moves to reach that position, maybe the move order in the actual game was a little bit different. In the position that you see I played d4.